harinris_Homework2
Harin Rishabh
2024-02-22
Chapter 1 - R Commands
Loading library
library(TSA)##
## Attaching package: 'TSA'## The following objects are masked from 'package:stats':
##
## acf, arima## The following object is masked from 'package:utils':
##
## tarExhibit 1.1
data(larain)
plot(larain,ylab='Inches',xlab='Year',type='o')
Exhibit 1.2
plot(y=larain,x=zlag(larain),ylab='Inches',xlab='Previous Year Inches')
Exhibit 1.3
data(color)
plot(color,ylab='Color Property',xlab='Batch',type='o')
Exhibit 1.4
plot(y=color,x=zlag(color),ylab='Color Property',
xlab='Prevous Batch Color Property')
Exhibit 1.5
data(hare)
plot(hare,ylab='Abundance',xlab='Year',type='o')
Exhibit 1.6
plot(y=hare,x=zlag(hare),ylab='Abundance',xlab='Previous Year Abundance')
Exhibit 1.7
data(tempdub)
plot(tempdub,ylab='Temperature',type='o')
Exhibit 1.8
data(oilfilters)
plot(oilfilters,type='o',ylab='Sales')
Exhibit 1.9
plot(oilfilters,type='l',ylab='Sales')
Month=c("J","A","S","O","N","D","J","F","M","A","M","J")
points(oilfilters,pch=Month)
Alternatively, the exhibit can be reproduced by the following commands
plot(oilfilters,type='l',ylab='Sales')
points(y=oilfilters,x=time(oilfilters),pch=as.vector(season(oilfilters)))
Chapter 2 - R Commands
Exhibit 2.1
rwalk contains a simulated random walk
data(rwalk)
plot(rwalk,type='o',ylab='Random Walk')
R code for simulating a random walk with, say 60, independant standard normal errors
n=60
set.seed(12345)
sim.random.walk=ts(cumsum(rnorm(n)),freq=1,start=1)
plot(sim.random.walk,type='o',ylab='Another Random Walk')
Chapter 3 - R Commands
Exhibit 3.1
time(rwalk) yields a time series of the time epoches when the random walk was sampled.
data(rwalk)
model1=lm(rwalk~time(rwalk))
summary(model1)##
## Call:
## lm(formula = rwalk ~ time(rwalk))
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.70045 -0.79782 0.06391 0.63064 2.22128
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.007888 0.297245 -3.391 0.00126 **
## time(rwalk) 0.134087 0.008475 15.822 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.137 on 58 degrees of freedom
## Multiple R-squared: 0.8119, Adjusted R-squared: 0.8086
## F-statistic: 250.3 on 1 and 58 DF, p-value: < 2.2e-16Exhibit 3.2
rwalk contains a simulated random walk
plot(rwalk,type='o',ylab='y')
abline(model1) # add the fitted least squares line
Exhibit 3.3
season(tempdub) creates a vector of the month index of the data as a factor
data(tempdub)
month.=season(tempdub) # the period sign is included to make the printout from
# the commands two line below clearer; ditto below.
model2=lm(tempdub~month.-1) # -1 removes the intercept term
summary(model2)##
## Call:
## lm(formula = tempdub ~ month. - 1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.2750 -2.2479 0.1125 1.8896 9.8250
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## month.January 16.608 0.987 16.83 <2e-16 ***
## month.February 20.650 0.987 20.92 <2e-16 ***
## month.March 32.475 0.987 32.90 <2e-16 ***
## month.April 46.525 0.987 47.14 <2e-16 ***
## month.May 58.092 0.987 58.86 <2e-16 ***
## month.June 67.500 0.987 68.39 <2e-16 ***
## month.July 71.717 0.987 72.66 <2e-16 ***
## month.August 69.333 0.987 70.25 <2e-16 ***
## month.September 61.025 0.987 61.83 <2e-16 ***
## month.October 50.975 0.987 51.65 <2e-16 ***
## month.November 36.650 0.987 37.13 <2e-16 ***
## month.December 23.642 0.987 23.95 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.419 on 132 degrees of freedom
## Multiple R-squared: 0.9957, Adjusted R-squared: 0.9953
## F-statistic: 2569 on 12 and 132 DF, p-value: < 2.2e-16Exhibit 3.4
model3=lm(tempdub~month.) # intercept is automatically included so one month (Jan) is dropped
summary(model3)##
## Call:
## lm(formula = tempdub ~ month.)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.2750 -2.2479 0.1125 1.8896 9.8250
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 16.608 0.987 16.828 < 2e-16 ***
## month.February 4.042 1.396 2.896 0.00443 **
## month.March 15.867 1.396 11.368 < 2e-16 ***
## month.April 29.917 1.396 21.434 < 2e-16 ***
## month.May 41.483 1.396 29.721 < 2e-16 ***
## month.June 50.892 1.396 36.461 < 2e-16 ***
## month.July 55.108 1.396 39.482 < 2e-16 ***
## month.August 52.725 1.396 37.775 < 2e-16 ***
## month.September 44.417 1.396 31.822 < 2e-16 ***
## month.October 34.367 1.396 24.622 < 2e-16 ***
## month.November 20.042 1.396 14.359 < 2e-16 ***
## month.December 7.033 1.396 5.039 1.51e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.419 on 132 degrees of freedom
## Multiple R-squared: 0.9712, Adjusted R-squared: 0.9688
## F-statistic: 405.1 on 11 and 132 DF, p-value: < 2.2e-16Exhibit 3.5
first creates the first pair of harmonic functions and then fit the model
har.=harmonic(tempdub,1)
model4=lm(tempdub~har.)
summary(model4)##
## Call:
## lm(formula = tempdub ~ har.)
##
## Residuals:
## Min 1Q Median 3Q Max
## -11.1580 -2.2756 -0.1457 2.3754 11.2671
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 46.2660 0.3088 149.816 < 2e-16 ***
## har.cos(2*pi*t) -26.7079 0.4367 -61.154 < 2e-16 ***
## har.sin(2*pi*t) -2.1697 0.4367 -4.968 1.93e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.706 on 141 degrees of freedom
## Multiple R-squared: 0.9639, Adjusted R-squared: 0.9634
## F-statistic: 1882 on 2 and 141 DF, p-value: < 2.2e-16Exhibit 3.6
plot(ts(fitted(model4),freq=12,start=c(1964,1)),ylab='Temperature',type='l',
ylim=range(c(fitted(model4),tempdub))) # the ylim option ensures that the
# y axis has a range that fits the raw data and the fitted values
points(tempdub)
Exhibit 3.7
data(rwalk)
model1=lm(rwalk~time(rwalk))
summary(model1)##
## Call:
## lm(formula = rwalk ~ time(rwalk))
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.70045 -0.79782 0.06391 0.63064 2.22128
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.007888 0.297245 -3.391 0.00126 **
## time(rwalk) 0.134087 0.008475 15.822 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.137 on 58 degrees of freedom
## Multiple R-squared: 0.8119, Adjusted R-squared: 0.8086
## F-statistic: 250.3 on 1 and 58 DF, p-value: < 2.2e-16Exhibit 3.8
plot(y=rstudent(model3),x=as.vector(time(tempdub)),xlab='Time',
ylab='Standardized Residuals',type='o')
Exhibit 3.9
plot(y=rstudent(model3),x=as.vector(time(tempdub)),xlab='Time',
ylab='Standardized Residuals',type='l')
points(y=rstudent(model3),x=as.vector(time(tempdub)),
pch=as.vector(season(tempdub)))
Exhibit 3.10
plot(y=rstudent(model3),x=as.vector(fitted(model3)),xlab='Fitted Trend Values',
ylab='Standardized Residuals',type="n")
points(y=rstudent(model3),x=as.vector(fitted(model3)),
pch=as.vector(season(tempdub)))
Exhibit 3.11
hist(rstudent(model3),xlab='Standardized Residuals',main='')
Exhibit 3.12
qqnorm(rstudent(model3),main='')
Exhibit 3.13
acf(rstudent(model3),main='')
Exhibit 3.14
plot(y=rstudent(model1),x=as.vector(time(rwalk)),ylab='Standardized Residuals',
xlab='Time',type='o')
Exhibit 3.15
plot(y=rstudent(model1),x=fitted(model1),ylab='Standardized Residuals',
xlab='Fitted Trend Values',type='p')
Exhibit 3.16
acf(rstudent(model1),main='')
Exercise 1
1.3
Answer - The plots are random.
plot(as.ts(rnorm(48)))
plot(as.ts(rnorm(48)))
plot(as.ts(rnorm(48)))
plot(as.ts(rnorm(48)))
1.4
Answer - Plots are random and non-normal.
plot(as.ts(rchisq(48, 2)))
plot(as.ts(rchisq(48, 2)))
plot(as.ts(rchisq(48, 2)))
plot(as.ts(rchisq(48, 2)))
1.5
Answer - Plots look to be random and non-normal.
plot(as.ts(rt(48, 5)))
plot(as.ts(rt(48, 5)))
plot(as.ts(rt(48, 5)))
plot(as.ts(rt(48, 5)))